Question 1 of 40
Use Cramer’s Rule to solve the following system.
3x – 4y = 4
2x + 2y = 12
A. {(3, 1)}
B. {(4, 2)}
C. {(5, 1)}
D. {(2, 1)}
Question 2 of 40
Use Gaussian elimination to find the complete solution to each system.
x1 + 4×2 + 3×3 – 6×4 = 5
x1 + 3×2 + x3 – 4×4 = 3
2×1 + 8×2 + 7×3 – 5×4 = 11
2×1 + 5×2 – 6×4 = 4
A. {(-47t + 4, 12t, 7t + 1, t)}
B. {(-37t + 2, 16t, -7t + 1, t)}
C. {(-35t + 3, 16t, -6t + 1, t)}
D. {(-27t + 2, 17t, -7t + 1, t)}
Question 3 of 40
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y – z = -2
2x – y + z = 5
-x + 2y + 2z = 1
A. {(0, -1, -2)}
B. {(2, 0, 2)}
C. {(1, -1, 2)}
D. {(4, -1, 3)}
Question 4 of 40
Use Gaussian elimination to find the complete solution to each system.
x – 3y + z = 1
-2x + y + 3z = -7
x – 4y + 2z = 0
A. {(2t + 4, t + 1, t)}
B. {(2t + 5, t + 2, t)}
C. {(1t + 3, t + 2, t)}
D. {(3t + 3, t + 1, t)}
Question 5 of 40
Give the order of the following matrix; if A = [aij], identify a32 and a23.
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1 |
-5 1/2 |
∏ 11 |
e -1/5 |
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A. 3 * 4; a32 = 1/45; a23 = 6
B. 3 * 4; a32 = 1/2; a23 = -6
C. 3 * 2; a32 = 1/3; a23 = -5
D. 2 * 3; a32 = 1/4; a23 = 4
Question 6 of 40
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
3×1 + 5×2 – 8×3 + 5×4 = -8
x1 + 2×2 – 3×3 + x4 = -7
2×1 + 3×2 – 7×3 + 3×4 = -11
4×1 + 8×2 – 10×3+ 7×4 = -10
A. {(1, -5, 3, 4)}
B. {(2, -1, 3, 5)}
C. {(1, 2, 3, 3)}
D. {(2, -2, 3, 4)}
Question 7 of 40
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + 2y = z – 1
x = 4 + y – z
x + y – 3z = -2
A. {(3, -1, 0)}
B. {(2, -1, 0)}
C. {(3, -2, 1)}
D. {(2, -1, 1)}
Question 8 of 40
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + 3y = 0
x + y + z = 1
3x – y – z = 11
A. {(3, -1, -1)}
B. {(2, -3, -1)}
C. {(2, -2, -4)}
D. {(2, 0, -1)}
Question 9 of 40
Use Cramer’s Rule to solve the following system.
x + 2y + 2z = 5
2x + 4y + 7z = 19
-2x – 5y – 2z = 8
A. {(33, -11, 4)}
B. {(13, 12, -3)}
C. {(23, -12, 3)}
D. {(13, -14, 3)}
Question 10 of 40
Use Gaussian elimination to find the complete solution to the following system of equations, or show that none exists.
2w + x – y = 3
w – 3x + 2y = -4
3w + x – 3y + z = 1
w + 2x – 4y – z = -2
A. {(1, 3, 2, 1)}
B. {(1, 4, 3, -1)}
C. {(1, 5, 1, 1)}
D. {(-1, 2, -2, 1)}
Solve the system using the inverse that is given for the coefficient matrix.
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2x + 6y + 6z = 8 |
The inverse of:
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2 2 2 |
6 7 7 |
6 6 7 |
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is
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7/2 -1 0 |
0 1 -1 |
-3 0 1 |
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A. {(1, 2, -1)}
B. {(2, 1, -1)}
C. {(1, 2, 0)}
D. {(1, 3, -1)}
Question 12 of 40
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
2x – y – z = 4
x + y – 5z = -4
x – 2y = 4
A. {(2, -1, 1)}
B. {(-2, -3, 0)}
C. {(3, -1, 2)}
D. {(3, -1, 0)}
Question 13 of 40
Use Cramer’s Rule to solve the following system.
2x = 3y + 2
5x = 51 – 4y
A. {(8, 2)}
B. {(3, -4)}
C. {(2, 5)}
D. {(7, 4)}
Question 14 of 40
Use Cramer’s Rule to solve the following system.
12x + 3y = 15
2x – 3y = 13
A. {(2, -3)}
B. {(1, 3)}
C. {(3, -5)}
D. {(1, -7)}
Question 15 of 40
Use Cramer’s Rule to solve the following system.
x + y + z = 0
2x – y + z = -1
-x + 3y – z = -8
A. {(-1, -3, 7)}
B. {(-6, -2, 4)}
C. {(-5, -2, 7)}
D. {(-4, -1, 7)}
Question 16 of 40
Use Cramer’s Rule to solve the following system.
4x – 5y – 6z = -1
x – 2y – 5z = -12
2x – y = 7
A. {(2, -3, 4)}
B. {(5, -7, 4)}
C. {(3, -3, 3)}
D. {(1, -3, 5)}
Find the products AB and BA to determine whether B is the multiplicative inverse of A.
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A = |
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0 0 1 |
1 0 0 |
0 1 0 |
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B = |
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0 1 0 |
0 0 1 |
1 0 0 |
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A. AB = I; BA = I3; B = A
B. AB = I3; BA = I3; B = A-1
C. AB = I; AB = I3; B = A-1
D. AB = I3; BA = I3; A = B-1
Question 18 of 40
Use Cramer’s Rule to solve the following system.
x + 2y = 3
3x – 4y = 4
A. {(3, 1/5)}
B. {(5, 1/3)}
C. {(1, 1/2)}
D. {(2, 1/2)}
Question 19 of 40
Solve the following system of equations using matrices. Use Gaussian elimination with back substitution or Gauss-Jordan elimination.
x + y + z = 4
x – y – z = 0
x – y + z = 2
A. {(3, 1, 0)}
B. {(2, 1, 1)}
C. {(4, 2, 1)}
D. {(2, 1, 0)}
Question 20 of 40
Use Cramer’s Rule to solve the following system.
4x – 5y = 17
2x + 3y = 3
A. {(3, -1)}
B. {(2, -1)}
C. {(3, -7)}
Question 21 of 40
Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola.
x2 – 2x – 4y + 9 = 0
A. (x – 4)2 = 4(y – 2); vertex: (1, 4); focus: (1, 3) ; directrix: y = 1
B. (x – 2)2 = 4(y – 3); vertex: (1, 2); focus: (1, 3) ; directrix: y = 3
C. (x – 1)2 = 4(y – 2); vertex: (1, 2); focus: (1, 3) ; directrix: y = 1
D. (x – 1)2 = 2(y – 2); vertex: (1, 3); focus: (1, 2) ; directrix: y = 5
Question 22 of 40
Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (0, -4), (0, 4)
Vertices: (0, -7), (0, 7)
A. x2/43 + y2/28 = 1
B. x2/33 + y2/49 = 1
C. x2/53 + y2/21 = 1
D. x2/13 + y2/39 = 1
Question 23 of 40
Convert each equation to standard form by completing the square on x and y.
9×2 + 25y2 – 36x + 50y – 164 = 0
A. (x – 2)2/25 + (y + 1)2/9 = 1
B. (x – 2)2/24 + (y + 1)2/36 = 1
C. (x – 2)2/35 + (y + 1)2/25 = 1
D. (x – 2)2/22 + (y + 1)2/50 = 1
Question 24 of 40
Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (-2, 0), (2, 0)
Y-intercepts: -3 and 3
A. x2/23 + y2/6 = 1
B. x2/24 + y2/2 = 1
C. x2/13 + y2/9 = 1
D. x2/28 + y2/19 = 1
Question 25 of 40
Locate the foci and find the equations of the asymptotes.
x2/9 – y2/25 = 1
A. Foci: ({±√36, 0) ;asymptotes: y = ±5/3x
B. Foci: ({±√38, 0) ;asymptotes: y = ±5/3x
C. Foci: ({±√34, 0) ;asymptotes: y = ±5/3x
D. Foci: ({±√54, 0) ;asymptotes: y = ±6/3x
Question 26 of 40
Find the vertex, focus, and directrix of each parabola with the given equation.
(x – 2)2 = 8(y – 1)
A. Vertex: (3, 1); focus: (1, 3); directrix: y = -1
B. Vertex: (2, 1); focus: (2, 3); directrix: y = -1
C. Vertex: (1, 1); focus: (2, 4); directrix: y = -1
D. Vertex: (2, 3); focus: (4, 3); directrix: y = -1
Question 27 of 40
Find the vertices and locate the foci of each hyperbola with the given equation.
y2/4 – x2/1 = 1
A. Vertices at (0, 5) and (0, -5); foci at (0, 14) and (0, -14)
B. Vertices at (0, 6) and (0, -6); foci at (0, 13) and (0, -13)
C. Vertices at (0, 2) and (0, -2); foci at (0, √5) and (0, -√5)
D. Vertices at (0, 1) and (0, -1); foci at (0, 12) and (0, -12)
Question 28 of 40
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Center: (4, -2)
Focus: (7, -2)
Vertex: (6, -2)
A. (x – 4)2/4 – (y + 2)2/5 = 1
B. (x – 4)2/7 – (y + 2)2/6 = 1
C. (x – 4)2/2 – (y + 2)2/6 = 1
D. (x – 4)2/3 – (y + 2)2/4 = 1
Question 29 of 40
Locate the foci and find the equations of the asymptotes.
4y2 – x2 = 1
A. (0, ±√4/2); asymptotes: y = ±1/3x
B. (0, ±√5/2); asymptotes: y = ±1/2x
C. (0, ±√5/4); asymptotes: y = ±1/3x
D. (0, ±√5/3); asymptotes: y = ±1/2x
Question 30 of 40
Find the standard form of the equation of the following ellipse satisfying the given conditions.
Foci: (-5, 0), (5, 0)
Vertices: (-8, 0), (8, 0)
A. x2/49 + y2/ 25 = 1
B. x2/64 + y2/39 = 1
C. x2/56 + y2/29 = 1
D. x2/36 + y2/27 = 1
Question 31 of 40
Locate the foci of the ellipse of the following equation.
25×2 + 4y2 = 100
A. Foci at (1, -√11) and (1, √11)
B. Foci at (0, -√25) and (0, √25)
C. Foci at (0, -√22) and (0, √22)
D. Foci at (0, -√21) and (0, √21)
Question 32 of 40
Find the standard form of the equation of each hyperbola satisfying the given conditions.
Foci: (0, -3), (0, 3)
Vertices: (0, -1), (0, 1)
A. y2 – x2/4 = 0
B. y2 – x2/8 = 1
C. y2 – x2/3 = 1
D. y2 – x2/2 = 0
Question 33 of 40
Locate the foci and find the equations of the asymptotes.
x2/100 – y2/64 = 1
A. Foci: ({= ±2√21, 0); asymptotes: y = ±2/5x
B. Foci: ({= ±2√31, 0); asymptotes: y = ±4/7x
C. Foci: ({= ±2√41, 0); asymptotes: y = ±4/7x
D. Foci: ({= ±2√41, 0); asymptotes: y = ±4/5x
Question 34 of 40
Find the focus and directrix of each parabola with the given equation.
x2 = -4y
A. Focus: (0, -1), directrix: y = 1
B. Focus: (0, -2), directrix: y = 1
C. Focus: (0, -4), directrix: y = 1
D. Focus: (0, -1), directrix: y = 2
Question 35 of 40
Convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola.
y2 – 2y + 12x – 35 = 0
A. (y – 2)2 = -10(x – 3); vertex: (3, 1); focus: (0, 1); directrix: x = 9
B. (y – 1)2 = -12(x – 3); vertex: (3, 1); focus: (0, 1); directrix: x = 6
C. (y – 5)2 = -14(x – 3); vertex: (2, 1); focus: (0, 1); directrix: x = 6
D. (y – 2)2 = -12(x – 3); vertex: (3, 1); focus: (0, 1); directrix: x = 8
Question 36 of 40
Find the standard form of the equation of the ellipse satisfying the given conditions.
Endpoints of major axis: (7, 9) and (7, 3)
Endpoints of minor axis: (5, 6) and (9, 6)
A. (x – 7)2/6 + (y – 6)2/7 = 1
B. (x – 7)2/5 + (y – 6)2/6 = 1
C. (x – 7)2/4 + (y – 6)2/9 = 1
D. (x – 5)2/4 + (y – 4)2/9 = 1
Question 37 of 40
Find the standard form of the equation of the ellipse satisfying the given conditions.
Major axis vertical with length = 10
Length of minor axis = 4
Center: (-2, 3)
A. (x + 2)2/4 + (y – 3)2/25 = 1
B. (x + 4)2/4 + (y – 2)2/25 = 1
C. (x + 3)2/4 + (y – 2)2/25 = 1
D. (x + 5)2/4 + (y – 2)2/25 = 1
Question 38 of 40
Find the solution set for each system by finding points of intersection.
x2 + y2 = 1
x2 + 9y = 9
A. {(0, -2), (0, 4)}
B. {(0, -2), (0, 1)}
C. {(0, -3), (0, 1)}
D. {(0, -1), (0, 1)}
None of the given answer choices is correct for the given system of equations.
However, if the second equation is supposed to be x2 + 9y2 = 9, then the answer would be D.
Question 39 of 40
Convert each equation to standard form by completing the square on x and y.
9×2 + 16y2 – 18x + 64y – 71 = 0
A. (x – 1)2/9 + (y + 2)2/18 = 1
B. (x – 1)2/18 + (y + 2)2/71 = 1
C. (x – 1)2/16 + (y + 2)2/9 = 1
D. (x – 1)2/64 + (y + 2)2/9 = 1
Question 40 of 40
Find the vertices and locate the foci of each hyperbola with the given equation.
x2/4 – y2/1 =1
A. Vertices at (2, 0) and (-2, 0); foci at (√5, 0) and (-√5, 0)
B. Vertices at (3, 0) and (-3 0); foci at (12, 0) and (-12, 0)
C. Vertices at (4, 0) and (-4, 0); foci at (16, 0) and (-16, 0)
D. Vertices at (5, 0) and (-5, 0); foci at (11, 0) and (-11, 0)
